Some remarks on Euler’s ϕ-function and some related problems
(1945)

Tools

"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."

ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c&gt; 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.

...d [29]), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime factors. Erdős =-=[6]-=- conjectured that for any constant c < 1 the inequality A(n) > n c holds infinitely often. Theorem 2. For some positive constant c there are infinitely many n such that both inequalities A(n) > n c an...

"... Given a fixed even integer k, we show that Schinzel’s hypothesis H implies that σ(n) = σ(n + k) infinitely often. We also discuss the case of odd k and the more general equation σα(n) = σα(n + k). 1 ..."

...Mientka and Vogt [7] asked for which values of k ≥ 1, σ(n) = σ(n + k) (2) has infinitely many solutions. Hunsucker, Nebb and Stearns [6] found at least two solutions of (2) for each k ≤ 5000. 1Erdős =-=[2]-=- made the much stronger conjecture that for every integer k ≥ 1 there is an n such that σ(n) = σ(n + 1) = · · · = σ(n + k), which would clearly imply that (2) has infinitely many solutions for each k....

...ound on � d∈A φ(d) relies on using Dressler’s result that only few integers are mapped to small values by φ. To state it formally, we define Mn = {x ∈ N|φ(x) ≤ n}. Dressler [3] proved (see also Erdös =-=[5]-=- and Bateman [2]): |Mn| = c ′ n + o(n) for c ′ = ζ(2)ζ(3) ζ(6) ≈ 1.9436. (22) It is clear that if A has the form Mn for some n,then � d∈Aφ(d) is minimised (with respect to a fixed |A|). Therefore choo...